The main aim of transformation (REF )
is that the description of dynamics
(originally provided by \(\mathfrak {h}\) )
is now shared by the two operators (viz., by \(H\)  and \(\Omega \)  or \(\Theta \) ).
In practical calculations, this makes the Dyson-inspired SP more flexible,
in principle at least.
Still, all of the mathematical operations
have to be performed,
exclusively, in the auxiliary
Hilbert space
\({\cal H}_{math}\) .
Although the latter space seems to
play just a technical role,
its key merit is that
it can be re-read,
after the mere ad hoc 
redefinition of the inner product
(see [1]}, [2]} or formula (REF ) below, with \(K=2\) )
as a new, unitarily equivalent representation
\({\cal H}_{phys}\)  of
the “old and missing” Hilbert space
\({\cal L}\) .
The conventional probabilistic interpretation
of the unitary quantum dynamics gets restored.
